Home

$x$ -intercepts of a Quadratic Function

The $x$ -intercepts of the function $f (x) = a x^{2} + b x + c, a \neq 0$ are the solutions of the quadratic equation $a x^{2} + b x + c = 0$ . The solutions of a quadratic equation of the form $a x^{2} + b x + c = 0$ are given by the quadratic formula.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Here, the expression $b^{2} - 4 a c$ is called the discriminant.

The discriminant can be used to confirm the number of $x$ -intercepts and the type of solutions of the quadratic equation.

Value of Discriminant	Type and Number of $x$ -intercepts (roots)	Example of Graph of Related Function
$b^{2} - 4 a c > 0;$ $b^{2} - 4 a c$ is a perfect square	$2$ real and rational	$f (x) = x^{2} - x - 6$
$b^{2} - 4 a c > 0;$ $b^{2} - 4 a c$ is not a perfect square	$2$ real and irrational	$f (x) = x^{2} - 3$
$b^{2} - 4 a c = 0$	$1$ real	$f (x) = {(x - 2)}^{2}$
$b^{2} - 4 a c < 0$	No $x$ -intercepts	$f (x) = x^{2} - 4 x + 7$

x -intercepts of a Quadratic Function

$x$ -intercepts of a Quadratic Function